Calculus question: What and why does the derivative have to relate to integral?

well im not sure i fully understand ur question so ill just explain some fundamental concepts and relationships between the two that should suffice for earlier calculus classes...

Derivative- tells u the slope at a given point on a line (I realize this applies only to the 1st Abercrombie derivative)
Integral- tells u the area enclosed between a line (ex. y=3x+6), an axis (usually the x axis), and provided limits. for example the integral for y=1 given 0 and 1 as limits is 1 since this is just the area of the unit square whose left corner is at the origin (0,0).

the relationship between the two is that they are practically opposites of eachother. Put simply, if you take the integral of an equation and then take the derivative of your result u will end up with your original equation. I can go much further in depth on this topic, however, I hope this gets the basic idea across.